Graph coloring On DNA && RNA At ﬁrst glance Graph theory may seem to be a rather abstract and theoretical mathematical ﬁeld that came into existence when Leonhard Euler solved a much celebrated problem known as the K¨onigsberg Bridges Problem in 1736. Since then it has been exploited for the solution of numerous problems and to this day practical aspects of graph theory have found a wide range of applications. We will present the application of graph theory coloring on DNA & RNA. The aim of our work is to find an algorithm to support the design of RNA molecules capable of forming two or more alternative metastable structures,DNA//////////. This required to create a logical information model, thus isolating relevant aspects of the biological problem and incorporating these into a graph-based mathematical model. So algorithm should reduces the problem by vertex coloring. Graph Coloring : A graph coloring is a consecutive assignment of “colors” to certain objects in a graph: these objects can be vertices, edges or a mixture of all of them. Most important for our purposes is the vertex coloring, usually assuming that no two adjacent vertices are allowed to be assigned the same color. A coloring of the graph G with k colors is called a k − coloring and is a function f : V (G)− > 1, 2, ..., k, such that no edge e = (u, v) has f (u) = f (v).